A005129 Theta series of {E_6}* lattice.

1, 0, 54, 72, 0, 432, 270, 0, 918, 720, 0, 2160, 936, 0, 2700, 2160, 0, 5184, 2214, 0, 5616, 3600, 0, 9504, 4590, 0, 9180, 6552, 0, 15120, 5184, 0, 14742, 10800, 0, 21600, 9360, 0, 19548, 12240, 0, 30240, 13500, 0, 28080, 17712, 0, 39744, 14760, 0, 32454

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

Comments

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

Examples

			G.f. = 1 + 54*x^2 + 72*x^3 + 432*x^5 + 270*x^6 + 918*x^8 + 720*x^9 + 2160*x^11 + ...
		

References

• J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 127.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
• G. Nebe and N. J. A. Sloane, Home page for this lattice

Crossrefs

Cf. A004007 (E_6).

Programs

Magma

A := Basis( ModularForms( Gamma1(3), 3), 51); A[1]; /* Michael Somos, Dec 28 2015 */

Mathematica

a[n_] := Module[{A = x*O[x]^n}, SeriesCoefficient[(QPochhammer[x+A]^3 / QPochhammer[x^3+A])^3 + 9*x*(QPochhammer[x^3+A]^3 / QPochhammer[x+A])^3, {x, 0, n}]]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Nov 05 2015, adapted from 1st PARI script *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ q]^3 / QPochhammer[ q^3])^3 + 9 q (QPochhammer[ q^3]^3 /QPochhammer[ q])^3, {q, 0, n}]; Table[a[n], {n, 0, 80}] (* Michael Somos, Dec 28 2015 *)

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^3 / eta(x^3 + A))^3 + 9 * x * (eta(x^3 + A)^3 / eta(x + A))^3, n))}; /* Michael Somos, Feb 28 2012 */

PARI

{a(n) = my(A, a1, p3); if( n<0, 0, A = x * O(x^n); a1 = sum( k=1, n, 6 * sumdiv(k, d, kronecker( d, 3)) * x^k, 1 + A); p3 = sum( k=1, n\3, -24 * sigma(k) * x^(3*k), 1 + A); polcoeff( (a1^3 + a1 * p3 - 4 * x * a1') / 2, n))}; /* Michael Somos, Feb 28 2012 */

Formula

Expansion of b(q)^3 + c(q)^3 / 3 in power of q where b(), c() are cubic AGM theta functions.

Expansion of (eta(q)^3 / eta(q^3))^3 + 9 * (eta(q^3)^3 / eta(q))^3 in powers of q.